An Infinite Genus Mapping Class Group and Stable Cohomology

Abstract

We exhibit a finitely generated group ${\mathcal {M}}$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface ${\mathcal {S}_\infty}$ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of ${\mathcal {M}}$ into the restricted symplectic group ${{\rm Sp_{\rm res}}(\mathcal {H}_r)}$ of the real Hilbert space generated by the homology classes of non-separating circles on ${\mathcal {S}_\infty}$ , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in ${H^2(\mathcal {M}, \mathbb {Z})}$ is the pull-back of the Pressley-Segal class on the restricted linear group ${{\rm GL_{\rm res}}(\mathcal {H})}$ via the inclusion ${{\rm Sp_{\rm res}}(\mathcal {H}_r) \subset {\rm GL_{\rm res}}(\mathcal {H})}$ .